Rolling two fair dices $A$ and $B$ and counting the number of odds where $B$ is rolled once more than $A$

Question

Suppose $A$ and $B$ each roll a fair six sided dice $101$, $100$ times respectively. If we count the number of odds and evens in each sequence, what is the probability that $A$ has more number of odds than $B$ does?

I think the probability is $1/2$ if we roll both $A$ and $B$ the same number of times. But how does the probability change if we roll $A$ one more time?